Nonlinear Stress-strain Relationship Research Articles

DEM exploration of the effect of particle shape on particle breakage in granular assemblies

Abstract The effect of particle shape on particle breakage is investigated through triaxial tests on sand assemblies mixed with non-spherical agglomerates using the discrete element method. Mixed assembly is generated in terms of cumulative distribution of sphericity of real sands. Comparable macro-mechanical results are found between DEM and experimental observations, including non-linear stress-strain relationship, peak deviatoric stress, and peak friction angle. More broken bonds and agglomerates numbers are generated in mixed assembly than that in spherical assembly. The hyperbolic relationship between particle breakage and energy input is found to have relevance to particle shape, and the fitting result of mixed assembly is in better accordance with real sands.

Dislocation-controlled microscopic mechanical phenomena in single crystal silicon under bending stress at room temperature

Silicon is widely used within energy, electro-mechanical, environmental devices by nanostructural control. As silicon parts constitute structural components whose size is ever decreasing, it is critical to understand the mechanical properties of single crystal silicon from precise measurements of load and displacement using microscopic sample in sub-micron and macroscopic scales. Here, the mechanical properties of single crystal silicon were precisely evaluated by bending tests at room temperature using microcantilever beam specimens having a several micron size. The microcantilever beam specimens were prepared using a focused ion beam technique, followed by loading the tip of the specimens. The smaller specimens deformed nonlinearly and then fractured. The unloaded specimen after nonlinear deformation showed permanent strain and many dislocations close to the region where high tensile stress was applied. This means that the nonlinear stress–strain relationship in the very high bending stress is determined by plastic deformation controlled by dislocation despite occurring at room temperature. The bending strength increased with a decrease in specimen size, and the smallest specimens had close to ideal strength. The size of the region where the dislocations accumulated in high density corresponded to the flaw size estimated from the fracture mechanics. This means that the bending strength of the microcantilever beam specimens of silicon is dominated by newly generated defects resulting from dislocations; in other words, the size effect of bending strength of silicon at the micrometer scale is controlled by the accumulation of newly formed dislocations because the dense dislocation region should be lower in a smaller-sized specimen.

Material characterisation of additively manufactured elastomers at different strain rates and build orientations

Material jetting, particularly PolyJet technology, is an additive manufacturing (AM) process which has introduced novel flexible elastomers used in bio-inspired soft robots, compliant structures and dampers. Finite Element Analysis (FEA) is a key tool for the development of such applications, which requires comprehensive material characterisation utilising advanced material models. However, in contrast to conventional rubbers, PolyJet elastomers have been less explored leading to a few material models with various limitations in fidelity. Therefore, one aim of this study was to characterise the mechanical response of the latest PolyJet elastomers, Agilus30 (A30) and Tango+ (T+), under large strain tension-compression and time-dependent high-frequency/relaxation loadings. Another aim was to calibrate a visco-hyperelastic material model to accurately predict these responses. Tensile, compressive, cyclic, dynamic mechanical analysis (DMA) and stress relaxation tests were carried out on pristine A30 and T+ samples. Quasi-static tension-compression tests were used to calibrate a 3-term Ogden hyperelastic model. Stress relaxation and DMA results were combined to determine the constants of a 5-term Prony series across a large window of relaxation time (10 μs–100 s). A numerical time-stepping scheme was employed to predict the visco-hyperelastic response of the 3D-printed elastomers at large strains and different strain rates. In addition, the anisotropy in the elastomers, which stemmed from build orientation, was explored. Highly nonlinear stress-strain relationships were observed in both elastomers, with a strong dependency on strain rate. Relaxation tests revealed that A30 and T+ elastomers relax to 50 % and 70 % of their peak stress values respectively in less than 20 s. The effect of orientation on the loading response was most pronounced with prints along the Z-direction, particularly at large strains and lower strain rates. Moreover, the visco-hyperelastic material model accurately predicted the large strain and time-dependent behaviour of both elastomers. Our findings will allow for the development of more accurate computational models of 3D-printed elastomers, which can be utilised for computer-aided design in novel applications requiring flexible or rate-sensitive AM materials.

Friction element for non-linear analysis of friction effect in externally prestressed beams

Analysis of externally prestressed beams involves particular difficulty related to slip between the external tendon and deviator. Slip without friction is usually assumed to simplify the problem. A concise friction element located at the angle bisector between external tendon and deviator is constructed, to simulate the slip with or without friction between the tendon and deviator. The non-linear stress–strain relationships of concrete, steel and external tendon are considered; a trilinear model of the bending moment–axial force–curvature of the beam section is adopted, so that the behaviour of the externally prestressed beam up to ultimate state can be investigated. Calculations regarding a simply supported beam and continuous beam for various factors are conducted, including different friction coefficients, different areas of external tendon and steel bar, different eccentricities, symmetric or asymmetric load patterns. Results indicate that, for the simply supported beam and continuous beam under symmetric load, the friction effect on load-carrying capacity is negligible; the friction effect on maximum prestress increment and deflection may not be negligible; the friction effect on minimum prestress increment is obvious. For a continuous beam under asymmetric load, the friction effect on load-carrying capacity, maximum and minimum prestress increments and deflection cannot be neglected.

Microstructure topology optimization by targeting prescribed nonlinear stress-strain relationships

In this paper, we proposed a new material design method by microstructure topology optimization. Novelty of the proposed method is to target the whole nonlinear volume-averaged effective stress-strain curve of microstructure representative volume element (RVE) rather than aiming specific values such as strength, stiffness or Poisson ratio. J2 plasticity model with a linear isotropic hardening model was chosen for local residuals. Global residuals are computed within nonlinear finite element framework for the topology optimization. Sensitivities of the objective function augmented with the residuals and adjoint response vectors with respect to design variables are derived with details and their numerical computational procedures were also presented. Microstructure topologies showing two different targeted stress-strain curves under uniaxial and biaxial loadings were obtained by using the method of moving asymptotes (MMA) optimization algorithm. Accuracy of the sensitivity computations was verified and numerical examples demonstrated a potential of the proposed method in applications to multiscale topology optimization.

The nonlinear orthotropic material model describing biaxial tensile behavior of PVC coated fabrics

The nonlinear orthotropic material model describing the biaxial tensile behavior of PVC coated fabrics is developed in this paper. Based on energy function, the general nonlinear stress-strain relationship is derived and the second-order and third-order nonlinear equations are proposed due to the in-plane stress-strain relationship assumption. The Membrane Structures Association of Japan (MSAJ) standard testing method for elastic constants of membrane materials is adopted and three types of PVC coated fabrics are tested under five different load ratios. The measured stress-strain data are used to calculate the coefficients of proposed nonlinear equations using the Least-square method of minimizing strain. Then the stress-strain curves of five different load ratios are compared among MSAJ suggested linear orthotropic model, proposed second-order nonlinear model, third-order nonlinear model, and the test data, also the 3D strain surfaces are plotted to make an intuitive comparison. Results show that both two proposed nonlinear models have a better representation of the test data than the linear model. Finally, considering the convenience and simplicity of the calculation, the second-order nonlinear model was integrated into the finite element software and further prove its adaptability. Thus it is recommended to apply the second-order nonlinear model to engineering design and numerical calculation.

Improved Junction Body Flow Modeling Through Data-Driven Symbolic Regression

A novel data-driven turbulence modeling framework is presented and applied to the problem of junction body flow. In particular, a symbolic regression approach is used to find nonlinear analytical expressions of the turbulent stress-strain coupling that are ready for implementation in computational fluid dynamics (CFD) solvers using Reynolds-averaged Navier-Stokes (RANS) closures. Results from baseline linear RANS closure calculations of a finite square-mounted cylinder with a Reynolds number of 11; 000, based on diameter and freestream velocity, are shown to considerably overpredict the separated flow region downstream of the square cylinder, mainly because of the failure of the model to accurately represent the complex vortex structure generated by the junction flow. In the present study, a symbolic regression tool built on a gene expression programming technique is used to find a nonlinear constitutive stress-strain relationship. In short, the algorithm finds the most appropriate linear combination of basis functions and spatially varying coefficients that approximate the turbulent stress tensor from high-fidelity data. Here, the high-fidelity data, or the so-called training data, were obtained from a hybrid RANS/Large Eddy Simulation (LES) calculation also developed with symbolic regression that showed excellent agreement with direct numerical simulation data. The present study, therefore, also demonstrates that training data required for RANS closure development can be obtained using computationally more affordable approaches, such as hybrid RANS/LES. A procedure is presented to evaluate which of the individual basis functions that are available for model development are most likely to produce a successful nonlinear closure. A new model is built using those basis functions only. This new model is then tested, i.e., an actual CFD calculation is performed, on the well-known periodic hills case and produces significantly better results than the linear baseline model, despite this test case being fundamentally different from the training case. Finally, the new model is shown to also improve predictive accuracy for a surface-mounted cube placed in a channel at a cube height Reynolds number of 40; 000 over traditional linear RANS closures.

Nonlinear vibration analysis of piezoelectric bending actuators: Theoretical and experimental studies

Piezoelectric bimorph actuators are used in a variety of applications, including micro positioning, vibration control, and micro robotics. The nature of the aforementioned applications calls for the dynamic characteristics identification of actuator at the embodiment design stage. For decades, many linear models have been presented to describe the dynamic behavior of this type of actuators; however, in many situations, such as resonant actuation, the piezoelectric actuators exhibit a softening nonlinear behavior; hence, an accurate dynamic model is demanded to properly predict the nonlinearity. In this study, first, the nonlinear stress–strain relationship of a piezoelectric material at high frequencies is modified. Then, based on the obtained constitutive equations and Euler–Bernoulli beam theory, a continuous nonlinear dynamic model for a piezoelectric bending actuator is presented. Next, the method of multiple scales is used to solve the discretized nonlinear differential equations. Finally, the results are compared with the ones obtained experimentally and nonlinear parameters are identified considering frequency response and phase response simultaneously. Also, in order to evaluate the accuracy of the proposed model, it is tested out of the identification range as well.

Pile under torque in nonlinear soils and soil-pile interfaces

This paper presents a new method for analyzing the nonlinear response of a single, vertical pile with the circular cross-section under torque in layered soils. The nonlinear stress-strain relationships of both soil-pile interface and soil are approximated by the hyperbolic model, whereas the pile material is elastic. The torsional spring stiffness of the soil-pile interface and the soil are determined by traditionally available methods. A four-node finite element model for the soil-pile interface is proposed to represent nonlinear behaviors of the soil-pile interface and the soil, separately. A new iterative scheme for nonlinear analysis of a single pile under torque is also developed that avoids having to solve a large number of simultaneous equations found in traditional solution schemes. The new solution method is based on the tangential stiffness of the soil-pile interface and soil springs, which are determined at each load step. From this solution scheme, an equivalent stiffness of the soil-pile-interface system of each pile element is calculated from the bottom element to the top element while torque and angle of twist are calculated from the top to the bottom elements. The solution gives the distribution of the angle of twist and torque along the pile, and the equivalent stiffness of the soil-pile system and torque-angle of twist curves at any depth. The solution method can be easily applied to the practice field in nonlinear analysis and in designing a single pile under torque in layered soils. The analysis results using the new solution scheme compared well with the results from other analytical methods studied by previous researchers. The proposed method is also used to predict the behavior of two full-scale piles under torques. The predictions are in good to excellent agreement with measurements.

Numerical Comparison of HPFRC and HPC Ribbed Slabs

The problem of limited resources is actual nowadays. One of the ways for solving this problem can be the use of high-performance fiber reinforced concrete as a structural material for elements subjected to the flexure. The dispersive reinforcement improves the properties of the concrete and allows reducing the required dimensions of cross-section and ratio of longitudinal reinforcement for members subjected to flexure. This research includes the numerical comparison of ultimate limit state and serviceability limit state of ribbed slabs from high-performance concrete (HPC) and high-performance fiber reinforced concrete (HPFRC) behavior, considering the different deformative properties of these materials. The effective use of these materials has been studied using a nonlinear finite element model. This model is created by using a beam type element with a multilayer cross-section, where each layer has a non-linear stress-strain relationship, according to the layer materials such as HPC, HPFRC, or steel bars. The finite element model used for the limited plates is validated. Results of experiment showed, that used calculation model describes with sufficient accuracy the behaviour of the ribbed slabs. The difference between design and experimentally determined breaking load is less than 6% for ribbed slab. The use of HPFRC for ribbed slabs with the spans from 6 to 12 m is justified so as using of HPFRC instead of HPC enables to increase 42-46% the intensity uniformly distributed load for the ribbed slabs with the same cross-sections and allowed deflections.

Nonlinear Elastic Effects on High-Strength Concrete Stress Invariants

Abstract Computational modeling of concrete behavior and most failure criteria for concrete strongly rely on the use of invariants of stress. These invariants allow a straightforward geometric interpretation of stress states and concrete failure surfaces in the three-dimensional Haigh-Westergaard stress space. Although second-order linear elastic relations have been traditionally applied to evaluate stresses in concrete, the magnitudes of concrete third-order parameters based on finite deformation theory—Murnaghan parameters l, m, n—are such that, if nonlinear stress-strain relations are considered, the stress invariants’ magnitudes are directly affected. In this work, acoustoelastic stress-strain relations for a general triaxial stress state and ultrasonically calculated second- and third-order elastic parameters are used in the evaluation of three high-strength concrete mixtures with compressive strengths: f′c = 43.0, 44.8, and 56.0 MPa (6.2, 6.5, and 8.1 ksi, respectively). The results demonstrate that nonlinear elastic effects introduce anisotropy when uniaxial stresses are applied and the magnitudes of concrete elastic and shear moduli change because of the applied stress field. When triaxial loads are considered, all stress invariants—first (I1), second (I2), and third (I3)—also undergo variations in their magnitudes. The stress-induced anisotropy due to uniaxial loading application and also the changes in the stress invariants’ magnitudes calls attention to the importance of considering nonlinear third-order effects in cement-based materials’ stress-strain relations and failure criteria.

Rectangular Section Model for Analysis of the Resistance of Eccentrically Compressed Masonry Structures

The paper presents section model for analysis of the resistance of masonry structures with rectangle cross-section subjected to the eccentric compression. The nonlinear stress-strain relationship for masonry in compression is assumed taking into account the effect of masonry softening. The formulae for normalized ultimate axial force and nmu and bending moment mmu are derived in the analytical form by integrating the cross-sectional equilibrium equations. The calculated results are presented in the form of interaction diagrams and compared with those based on the parabola-rectangle diagram for masonry in compression. It has been shown that the section resistance is strongly influenced by the masonry softening and depends on the limiting value εmu for masonry in compression.

A study on the nonlinear elastic behavior of ballistic gelatin using a rotational rheometer

The present work was focused on the study of large elastic behavior and constitutive modeling of ballistic gelatin. The simple shear tests were conducted using a rotational rheometer. It is observed that the mechanical parameters of the material should be analyzed directly by the data of rotational angle and torque measured by the rheometer but not the apparent shear stress and strain data provided by the rheometer directly. A hyperelastic constitutive model, a 2-term reduced polynomial strain energy potential, was adopted to describe the nonlinear stress-strain relationship. The material parameters were identified and the dependency of the parameters on the temperature was analyzed. It’s found that the modulus at small deformation decreases with increasing temperature following an approximately linear relationship; while, the modulus at large deformation showed a process of first increasing and then decreasing with increasing temperature.

An analytical method to predict the buckling and collapse behaviour of plates and stiffened panels under cyclic loading

The nonlinear stress-strain relationship of a thin plate or stiffened panel under in-plane load is represented by a load-shortening curve. The curves are used to evaluate the buckling and ultimate collapse behaviour of these structural elements, and furthermore forming the input data to analytical progressive collapse methods for large scale box girder structures such as ships. This paper develops a novel analytical method that predicts the load-shortening curve of plates and stiffened panels under cyclic in-plane load. This provides the framework to account for load reversals in an enhanced cyclic progressive collapse method. A parametric study using nonlinear finite element analysis is completed to investigate the characteristic behaviour of simply supported plates under cyclic compression and tension. The investigation covers a range of aspect ratios and slenderness ratios typical for ship-type structures. Single-cycle and ten-cycle loading protocols are applied, which demonstrate progressive reduction in strength and stiffness together with a response convergence after several cycles. An analytical method to predict multi-cycle load-shortening behaviour is then derived using a response and updating rule based on the observed characteristics from the parametric study. A validation of the analytical method is performed on a range of unstiffened plates and stiffened panels under various cyclic loading protocols. A good comparison with the results of finite element analysis is obtained, which confirms the validity of the proposed analytical method.

Effect of fiber length on the tensile strength of unidirectionally arrayed chopped strands

This study investigates the effect of fiber length on the tensile strength for discontinuous fiber composites. A composite with a staggered structure of fiber bundles were prepared by stacking prepreg sheets with unidirectionally arrayed chopped strands with a predetermined fiber length. The sheets were prepared by introducing fine slits at regular intervals along the fiber direction. The final failure is caused by an unstable growth of delamination from the tip of the bundles. The conventional linear fracture mechanics model cannot explain the reduction in the tensile strength as the fiber length shortens. Consequently, this paper proposes a novel fracture mechanics model that considers a non-linear stress–strain relation for the composite, which incorporates varying fiber length. The results confirm that this model can explain the relation between the tensile strength and fiber length very well, without any fitting parameters.

Mechanical characterisation of human and porcine scalp tissue at dynamic strain rates

Several biomedical applications require knowledge of the behaviour of the scalp, including skin grafting, skin expansion and head impact biomechanics. Scalp tissue exhibits a non-linear stress-strain relationship, anisotropy and its mechanical properties depend on strain rate. When modelling the behaviour of the scalp, all these factors should be considered in order to perform realistic simulations. Here, tensile tests at strain rates between 0.005 and 100 s−1 have been conducted on porcine and human scalp in order to investigate the non-linearity, anisotropy, and strain rate dependence of the scalp mechanical properties. The effect of the orientation of the sample with respect to the Skin Tension Lines (STLs) was considered during the test. The results showed that anisotropy is evident in the hyperelastic response at low strain rates (0.005 s−1) but not at higher strain rates (15-100 s−1). The mechanical properties of porcine scalp differ from human scalp. In particular, the elastic modulus and the Ultimate Tensile Strength (UTS) of the porcine scalp were found to be almost twice the values of the human scalp, whereas the stretch at failure was not found to be significantly different. An anisotropic hyperelastic model (Gasser-Ogden-Holzapfel) was used to model the quasi-static behaviour of the tissue, whereas three different isotropic hyperelastic models (Fung, Gent and Ogden) were used to model the behaviour of scalp tissue at higher strain rates. The experimental results outlined here have important implications for those wishing to model the mechanical behaviour of scalp tissue both under quasi-static and dynamic loading conditions.

Multipole borehole acoustic field in a transversely isotropic medium induced by stress

A borehole multipole acoustic field in a pre-stressed formation is investigated. The pre-stressed formation is modeled as a transversely isotropic medium induced by uniaxial stress. The formation is assumed to be isotropic in absence of any static stress and then becomes anisotropic due to the applied stress parallel to the borehole axis. The approximate equivalent elastic constants of the stress-induced anisotropic medium are derived from the theory of acoustoelasticity. The nonlinear static stress-strain relation is used for both small and large static deformations. This problem can be solved analytically because of uniformity of deformation induced by static stress applied parallel to the borehole axis. The stress effects on the velocity of guided waves and amplitude of waveforms excited by monopole, dipole, and quadrupole sources are investigated. Numerical results show that the velocities of guided waves increase with uniaxial stress. The uniaxial stress affects both amplitude and arrival time of the acoustic waves in the borehole. The integral amplitude of full waveforms varies almost in a parabolic manner with the increasing stress level and thus shows sensitivity to the uniaxial stress. This result may be helpful for remote stress measurements in boreholes.

A dual-pillar method for measurement of stress-strain response of material at microscale

A dual-pillar method is proposed to determine nonlinear stress-strain relation and mechanical properties of shape memory alloy at microscale. Two neighboring cuboidal micropillars of the same cross-sectional area but different heights are fabricated and compressed by flat-tip nanoindentation. An accurate stress-strain response of the middle uniformly deformed part of the long pillar is obtained by subtracting the measured displacement of the short pillar from that of the long pillar to remove the top and bottom effects. Finite element analysis confirmed the effectiveness of the developed dual-pillar method in removing the effects of substrate deformation and stress concentration in single-pillar compression. This method is also valid to other materials and to tensile loading condition.

Nonlinear Lamb wave for the evaluation of creep damage in modified 9Cr–1Mo steel

Higher harmonic generation using nonlinear ultrasonic is known to be very sensitive to microstructural degradation. This technique relies on nonlinear stress-strain relation i.e. nonlinear Hooke's law. The quantitative measurement to characterize the materials degradation is done by the nonlinear ultrasonic parameter, β. The objective of this work is to use nonlinear ultrasonic using Lamb wave to evaluate creep damage in modified 9Cr–1Mo steel. Relative change in β was seen to be dependent on the higher order elastic constants that evolve with microstructure as creep progresses as well as sub-structural changes. The extent of damage in the material due to creep exposure was evaluated by Kernel Average Misorientation (KAM) from EBSD and co-related with nonlinear acoustic parameter.

Static and dynamic flame model effects on thermal buckling: Fixed-roof tanks adjacent to an ethanol pool-fire

Storage tank fires (such as pool-fires) often occur in the petrochemical tank farms. Flame pulsation is an important characteristic of the turbulent flames observed in pool-fires. Current cylindrical solid flame models inadequately predict the thermal radiation of pool-fires because of the ignorance of the flame pulsation effect. In this study, the thermal buckling behavior and fire resistance of a fixed-roof Q345 steel tank with a stepped thickness exposed to a neighboring ethanol pool-fire based on the flame pulsation model is numerically investigated. The influence of smoke generated by the combustion process which can reduce thermal radiation fluxes is taken into account. Geometric and material nonlinearity which considers nonlinear strain-displacement and stress-strain relation respectively is used on finite element analysis. Results show that the thermal buckling mode of the cylindrical tank wall is elastic buckling, and the thermal buckling behavior is non-linear. The fire resistance of the fixed-roof steel tank increases significantly as the vertical fire location increases, and the fire resistance decreases with the burning tank diameter increasing. Moreover, the fire resistance of the fixed-roof steel tank for a cylinder-cone combined flame based on the flame pulsation model is larger than that for a cylindrical flame. The study can be used to optimize the steel structure design of oil tanks for resisting pool-fires, and thus reduce the loss caused by fire accidents in tank farms.